Computer Code for the Calculation of Thermal Neutron Absorption in Spherical and Cylindrical Neutron Sources

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Computer Code for the Calculation

of Thermal Neutron Absorption

in Spherical and Cylindrical

Neutron Sources — NATIONAL BUREAU OF STANDARDS

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NBS TECHNICAL NOTE 576 ISSUED MAY 1971

Nat. Bur. Stand. (U.S.), Tech. Note 576, 25 pages (May 1971) CODEN: NBTNA

Computer Code for the Calculation of Thermal Neutron Absorption

in Spherical and Cylindrical Neutron Sources

V. Spiegel, Jr. and W. M. Murphey

Nuclear Radiation Division Institute for Basic Standards National Bureau of Standards Washington, D.C. 20234

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NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402. (Order by SD Catalog No. C 13.46:576). Price 35 cents. Stock No. 0303 0851 Abstract

A computer code has been written in FORTRAN IV for the calculation of thermal neutron absorption in spherical and cylindrical neutron sources. The formalism of the calculation, the structure of the computer code, a listing of the code, and some sample results are presented. The com- parison of the results of this calculation to experiment appears elsewhere (l).

Key words: Manganous sulfate bath calibration of neutron sources; neutron; neutron standards.

11 Computer Code for the Calculation of Thermal Neutron Absorption in Spherical and Cylindrical Neutron Sources

V. Spiegel, Jr. and William M. Murphey

1. INTRODUCTION

This calculation has been carried out in connection with a program to reduce the uncertainties in the corrections applied to the manganous sulfate bath calibration of neutron sources (2,3). The correction considered here is to account for the reduction of the manganese activity due to the loss of thermalized neutrons absorbed in the neutron source itself.

The source may be composed of up to three cladding and one, possibly fissionable, core material. The calculation is carried out in a single interaction approximation, i.e., the effects of elastic and inelastic scattering of thermal neutrons are neglected. This approximation is adequate because the neutrons are in thermal equilibrium and because the correction which is applied to the calibration is small (typi- cally Y% or less). Two cases are available. The first is for a spherically symmetric source and the second is for a cylindrically symmetric source. Each consists of a core and up to three cladding layers. The thicknesses of the ends and side of a cladding cylinder may all be different. A measurement or knowledge of the thermal -neutron flux at the source location is required. The thermal -neutron flux is assumed to be isotropic, which enables one to carry out the computation as a sum of mono-directional fluxes from different directions. All integrations are performed with Weddle's formula (4).

2. DESCRIPTION OF THE CALCULATION

Part I. The probability of neutron loss for a given neutron direction and position.

If the source materials are labeled A, B, C, and D from the inside out, then the probability of a thermal neutron interacting in passing through the source is

2 £ - Saa(t)-Ebb(t)- cc(t)- dd(t) where a(t) is the thickness of "A" material for this particular direction and location of passing through the source, b(t) is the total thickness of "B" material, etc., and "t" denotes any particular path through the source. The E's are £ the appropriate macroscopic cross sections, a being the sum of

Present address is: Technical Analysis Division, Institute for Applied Technology, National Bureau of Standards, Washington, D.C 20234. the absorption and fission cross sections for the "A" material, 2 being the absorption cross section for the "B" material, etc. The probability of fission in the fissionable material is given by

E "E a(t) , , af , a w -(V^^V^^V^)}d c ^ . \ P (t) (l-e )(e ) f »_i£ (2) S a where d»(t) is the thickness of the "D" layer passed in this direction and location going from the outside into the MA" material, c'(t) the thickness of the "C" layer passed in going into the source, etc., and zaf is the macroscopic fission cross section of material "A".

Part II. Case 1. A spherically symmetric source.

In this case the probability of neutron disappearance for a neutron striking the source is given by R 2Tiy p(y) dy % (3)

TT R2 where R is the outer radius of the source and y is the perpendicular distance from the center of the source to the path through it. The probability of fission is similarly

,R 2TTy P (y) dy 'o f Pf « (4) 2 TTR

The probability of loss of neutrons from the bath per neutron striking the source is therefore = PL P-kPf (5) where k is the number of neutrons per fission interaction.

Part II. Case 2. A cylindrically symmetric source.

The notation for the dimensions of the source is given in Figure 1. Since an isotropic flux is assumed it is necessary only to consider monodirectional fluxes from an appropriate number of different directions. By symmetry it is most con- venient to use cylindrical coordinates. For convenience in normalization and avoidance of solid angle considerations an imaginary sphere is placed about the source and all neutrons striking this sphere are included in the integrals and probabilities. This procedure does not affect the results when they are finally expressed in terms of neutrons lost as a percent of the source sti'ength or in terras of a given thermal flux in the vicinity of the source.

Given that a neutron enters this imaginary sphere, let of transmission interaction the probability without be PT , probability of interaction be Pj, probability of absorption be PA and the probability of fission be Pf . The probability of neutron loss in the source PL If PL - Pj-kPf = l-PT -kPf (6) where k is the number of neutrons per fission. The program computes P and P by integration and p_ and P by P = 1- T f A T PT and P = Pj -P . A f

Part III. The integration and normalization.

Consider the cylindrical source surrounded with a sphere centered about the core material with radius, R, just large enough to enclose the outer cylinder as shown in Figure 1(a). The thermal -neutron flux is taken to be isotropic. Neutron flux is the number of neutrons per second entering a sphere of 1 square centimeter cross section. The number of neutrons per second entering the sphere surrounding the cylindrical source is the cross sectional area of the surrounding sphere times the thermal flux.

Since the geometry is symmetric about the x-axis, the probability for various results for an isotropic flux will equal the probabilities evaluated assuming a flux uniformly distri- buted in z, w, and cp but all paths constrained to lie in planes parallel to the x-y plane. The neutron paths are perpendicular to all points along the w-axis within the sphere and cp is the angle between the y and w axes. The probability of transmission of a neutron through the imaginary sphere averaged over all directions is thus R 2* r(z) = .2 Pm = dz dcp I* dw P (z,w,cp)/2tt(ttR ) (7) T J [ t -R b -r(z)

The probability of transmission, Pt (z,w,«p) is

-(2 a(z,w,cp)+S b(z,w,cp)+i: _c(z,w,cp)+Z d(z,w,cp)) a h c ad (z,w,cp)= b PT e (8) where the £'s are the macroscopic cross sections. Parameters are as in the spherical case and the a, b, c, and d are the thicknesses of the A, B, C, and D materials as seen at the particular z, w, and cp. For a transparent source the macroscopic cross sections are all zero and equation (7) integrates to unity. The fission probability is similarly

R 2tt r(z) 2 * dz dcp dw (z,w,cp)/2rT(nR Pp ^ ^ ^ P f ) (9) -R o -r(z)

-P -Y where P (z,w,cp) = (1-e )e f (^ afA a ).

In (z,w,cp). P f

= E a(z,w,cp) p a ,

-P 1-e is the probability for interaction in the "A" material,

Y - 2 b (z,w,cp)+S c (z,w,cp)+E d (z,w,cp) b 1 c i d 1 where b^(z,w,

3. DESCRIPTION OF THE CODE

3.1 Spherical Case Calculation

3.1.1 Organization of the Code

The program first reads the title card and input data, converts the source dimensions from inches to centimeters and then prints out an echo check of the title, input data and converted dimensions.

The approximate total number of subdivisions, NY, of the radius is specified in the input. Then each layer is subdivided in proportion to its thickness in the radial direction. The boundary between the central core and s first encapsulation or the boundary between any of the encapsulations are always made an endpoint of subdivision so that any discontinuities occasioned by a change in derivative at a boundary do not occur within an interval of integration.

The integration of Equation (4) then begins with the central core and continues to the outside radius, layer by layer. All neutron paths at a given radius are perpendicular to a cross section through the center of the source and are equal in length. Subroutine Grand calculates the interaction probability in the A, B, C, and D materials and the transmission probability along the path specified. The probability of interaction in each material and the transmission probability with its error are summed and normalized in the main program. The results are then printed below the echo printout of the input data.

3.1.2 Approximations in the Numerical Integration

The numerical integration was performed using Weddle ' Rule (4). If the core material has been divided into N subdivisions, Weddle 's Rule replaces the integrand of the integral by N fifth-degree parabolas and numerically integrates over that region. The accuracy of the integration and estimate of the error depend upon the size of the integration intervals. Convergence of the integration procedure was checked for a plutonium-beryllium source with encapsulations of tantalum and stainless steel by varying the number of subdivisions, NY, of the radius between 10 and 1,000 In Table 1 the calculated transmissions and estimated errors are shown together with the actual error based upon accepting the finest subdivision as the correct answer. The estimated and actual error appear to agree for 20 subdivisions of NY, but this degree of accuracy is certainly not necessary. The difference in net neutron loss calculated for NY equal 10 and 1000 is only 0.005$ out of 0.278$.

This program has been run in double precision, because it was created, for the most part, from the deck of cards to the cylindrical case, which had to be run in double precision. The number of additions involved in the transmission integral is approximately equal to 6 times the number of subdivisions, NY, of the radius specified in the input. Therefore no rounding error should be introduced by performing the program in single precision. The program takes about 3 seconds to compile and about 2 seconds to execute in the case of NY equal 100. It requires 1437 words of memory. 3.2 Cylindrical Case Calculation

3.2.1 Organization of the Code

Here too the title card and input data are read, the source dimensions are converted to centimeters, and an echo check of the input is printed. According to the number, NZ, of subdivisions in the Z- direction, specified in the input, each layer in the Z-direction is subdivided in proportion to its thickness. The subdivision of the angle of integration from zero to tt/2 is performed according to the number, NPHI, specified in the input. The subdivisions along the W-axis are set at the interface of any two materials in a plane parallel to the X-Y axis, distance Z above the origin. The D, C, and B materials in the negative W-direction, the core A material, and the B, C, and D materials in the positive W-direction are all divided into six subintervals. The limits of inte- gation for each of these subintervals along the W-axis are passed to Subroutine IN. It checks the length of the interval and will further subdivide it by 6 or 36, if it is more than 6 times the length of a subdivision in the Z -direction. It sums and weights the contributions to the transmission, the error for the transmission, and the interaction in the core material for each path through the source perpendicular to the W-axis between the specified limits of integration and returns the result to the main program. These in turn are summed and weighted according to the Weddle Rule for the integration over angle,

3.2.2 Approximations in the Numerical Integration

The comments about the use of the Weddle Rule in 3.1.2 also apply here. In this case we integrate over Z,

The convergence of the cylindrical program has been test- ed for two different plutonium-beryllium sources, one with a single nickel encapsulation and the other with encapsulations of tantalum and stainless steel. In Figure 2 are shown the calculated transmissions and estimated errors for the nickel encapsulated source for various specified subdivisions of the Z and

This program was affected by rounding errors when run in 'single precision by two different compilers. The results were only independent of choice of compiler, when run in double precision. The program takes about 5 seconds to compile and requires 2992 words of memory. Typical run times are listed in the above paragraph.

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4. LISTING OF THE PROGRAMS

4.1 SPHERICAL CASE

4.1.1 LIST OF THE PROGRAM

IMPLICIT DOUBLE PREC I S I ON ( A-H ,0-Z 1 DIMENSION V<6) .T< 14) .EH<6) , RZZ(5),KA(4) 2 DATA V/2. DO, 5. DO, 1. DO. 6. DO, 1. DO, 5. DO/, EH/ 1 . DO ,-6 . DO 3 ]15.D0,-2 0.D0.15.D0.-6.D0/,CC/l.D-8/ 4 1 FORMAT(13A6,A2/7G8.4,I4) 5 3 FORMATU2G6.3) 6

4 F0RMAT(1H1,13A6,A2/' SOURCE DlMENS IONS ' 25X ' / MACROSCOPIC CROSS SEC 7 1 '17X'/ INTEGRATION PARAMETERS'/' RA ='G14.6,' TB ='G14.6,' 8 1 / SIGAA='G14.6, ' AK ='G14.6.' / Ny ='16/' RB ='G14.6»« TC 9 1 ='G14.6,' / SIGAF='G14.6, « F ='G14.6» /' RC ='G1 10 14.6,' TD ='G14.6,' / SIGB ='G14.6» /' RD ='G 11 114.6.21X 1 / SIGC ='G14. 6.21X1 / NCY ='16/ 42X 12 1 ' / SIGD ='G14.6,21X' /' / 13 ] 11X • IN INCHES, ' 2X • / COVER / WALL 14

1 THICKNESS '/13X'OUTSIDE DI AMETER= • G12 . 6 15 1' / B'4XG12.6» /43X'/ C4XG12.6. 16 1 /43X'/ D'4XG12.6) 17 5 FORMATI3X/ 46H FOR A SINGLE NEUTRON STRIKING THE SOURCE.../ 18 120X35HTHE TRANSMISSION PROBABILITY IS 22XG16.8/ 19 120X35HTHE INTERACTION PROBABILITY IS 22XG16.8/ 20 120X35HTHE ABSORPTION PROBABILITY IS 22XG16.8/ 21 120X44HTHE 'A' MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 22 120X44HTHE 'B' MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 23 120X44HTHE 'C MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 24 120X44HTHE 'D' MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 25 120X57HTHE PROBABILITY FOR A FISSION ABSORPTION IS G16 26 1.8/16X' NEUTRONS OUT PER NEUTRON STRIKE IS ' 22XG16.8 27 1/20X'NEUTRONS LOST PER NEUTRON STRIKE '22XG16.8,' .' 28 1//38H THE CROSS SECTION OF THE SOURCE IS G13.8.' CM**2.' 29 1/3X'THE ERROR IN THE TRANSMISSION PROBABILITY IS ESTIMATED TO BE 30

l'G12.3. 'PERCENT. ' ) 31

7 FORMAT) 3X/ 44H FOR A FLUX OF ONE NEUTRON PER CM**2-SEC * ( G 12'. 5 . 32 141H NEUTRONS PER SEC) STRIKING THE SOURCE.../ 33 120X26HTHE TRANSMISSION IS G16.8.20H NEUTRONS PER SECOND/ 34 120X26HTHE INTERACTION IS G16.8.20H NEUTRONS PER SECOND/ 35 120X26HTHE ABSORPTION IS G16.8.20H NEUTRONS PER SECOND/ 36 120X26HTHE FISSION EMISSION IS G16.8,20H NEUTRONS PER SECOND/ 37 116X26HAND THE RESULTANT LOSS IS 4XG16.8,' NEUTRONS PER SECOND.') 38

8 FORMATOX/ 35H IF THE MEASURED THERMAL FLUX IS G12.4, ' PER CE 39 INT OF Q PER CM**2 THEN...'/ 40 120X26HTHE TRANSMISSION IS G16.8.14H PER CENT OF Q/ 41 120X26HTHE INTERACTION IS G16.8.14H PER CENT OF Q/ 42 120X26HTHE ABSORPTION IS G16.8.14H PER CENT OF Q/ 43 120X26HTHE FISSION EMISSION IS G16.8.14H PER CENT OF Q/ 44 116X26HAND THE RESULTANT LOSS IS 4XG16.8,' PER CENT OF Q.') 45 9 FORMAT) lX'THE NUMBER OF GRAND CALLS WAS' 18.' THE LARGEST SUBDIVIS 46 HON OF THE RADIUS WAS 'G16.8) 47 100 READ(5.1 IT.SIGAA.SIGAF, SIGB, SIGC. SIGD. AK,F, NY 48

PEADI5.3) DIA.TBI ,TCI .TDI 'OUTSIDE D I AMETER+SHELL THICKNESSES 49 IGRAND=0 "COUNTER RESET FOR NUMBER OF SUB GRAND CALLS 50 NCY=4 'NCY EQUALS THE NUMBER OF LAYERS 51

IF(TDI.LT.CC) NCY=3 • 3 LAYERS 52

IF(TCI.LT.CC) NCY=2 • 2 LAYERS 53

IF(TBI.LT.CC) NCY=1 ' 1 LAYER 54 TB=TBI*2.54D0 'THE INPUT DIMENSIONS WERE IN INCHES 55 TC=TCI*2.54D0 'F=THE NEUTRON FLUX IN PERCENT OF Q 56 TD=TDI*2.54D0 'T=TITLE. AK=NEUTRONS PER FISSION 57 RD=DIA*2.54D0/2.D0 'OUTSIDE RADIUS 58 RC = RD-TD 'FOR ONE LAYER A, B.C. AND D DIMENS. ARE EQUAL 59 RB=RC-TC 'FOR TWO LAYERS B.C. AND D DIMENSIONS ARE EQUAL 60 RA=RB-TB 'FOR THREE LAYERS C AND D DIMENSIONS ARE EQUAL 61 DY=0.D0 'LARGEST DIV. OF RADIUS. Y. IS MAX VALUE OF DA 62 SIGAT=SIGAF+SI6AA 'SIGAT=TOTAL CROSS SECTION, 'A' MATERIAL 63 i, )

WRITE(6»4)T,RA»TB,SIGAA ,AK,NY,RB,TC,SIGAF,F,RC,TD,SIGB»RD,SIGC» 64 1 NCY,SIGD,DIA,TBI ,TCI ,T DI 65 RZZ< 1 )=0.D0 66 RZZ(2)=RA 67 RZZ(3)=RB 68 RZZ(4)=RC 69 RZZ(5 )=RD 70 EKI=O.DO 71 STI=O.DO 72 SYD=O.DO 73 SYC=O.DO 74 SYB=0.D0 75 SYA=0.D0 76 ANY=NY/RD •NY=APPROXIMATE NO. OF Y INTERVALS 77 KA( 1)=RA*ANY+.5D0 'THIS IS TO SUBDIVI DE THE Y-LAYERS IN 78 KA(2)=(RB-RA)*ANY+.5D0 •PROPORTION TO THE THICKNESS OF EACH LAYER 79 KA( 3)=(RC-RB)*ANY+.5D0 •FROM ZERO TO RA, F ROM RA TO RB, FROM RB 80 KA(4)=(RD-RC)*ANY+.5D0 •TO RC, AND FROM RC TO RD. Y-INTEGRATION WILL 81 DO 281 IAR=1,NCY •ALWAYS OCCUR AT LA YER BOUNDARIES AND AVOID 82 IF(KA( IAR).LT.l )KA( IAR) =1 'DISCONTINUITIES 83

DA=(RZZ( IAR+1)-RZZ( IAR) )/<6.D0*KA( IAR) ) 84 IF(DA.GT.DY)DY=DA •LARGEST DIVISION F RADIUS, Y 85 NYP=6*KA( IAR)+1 'THE NUMBER OF PTS OF INTEGR. IN EACH LAYER 86 IVY = 2 •IVY= Y INTEGRATION WEDDLE RULE COEF INDICATOR 87 SPYT=0.D0 'Y PARTIAL SUM RESE T OF TRANSMISSION AND 88 SPYD=0.D0 •INTERACTION IN 'D • MATERIAL 89 SPYC=0.D0 •INTERACTION IN 'C • MATERIAL 90 SPYB=0.D0 'INTERACTION IN 'B • MATERIAL 91

SPYA=0.D0 •INTERACTION IN 'A • MATERIAL 92 EK=0.D0 'ERROR RESET FOR Y INTEGRAL IN DO LOOP 280 93 TEK=0.D0 •TEMPORARY EK 94 DO 280 1=1, NYP •BEGIN Y INTEG R.FR O-RA-RB-RC-RD 95 I Y=DA*DBLE ( -1 ) +RZZ ( I AR ) 96 PDY=0.D0 97 RCY=0.D0 98 RBY=0.D0 99 RAY=0.D0 100 GO TO (173,172,171,170) ,NCY 101 170 IF(RD.GT,Y)RDY=DSQRT(RD **2-Y**2 102 171 IF(RC.GT.Y) RCY=DSQRT(R C**2-Y**2) 'HIT C LAYER, IF THERE IS ONE 103 172 IF(RB.GT.Y) RBY=DSQRT(R B**2-Y**2) 'HIT B LAYER, IF THERE IS ONE 104 173 IF(RA.GT.Y) RAY=DSQRT(R A**2-Y**2) 'HIT A LAYER 105

CALL GRAND( Y,SPIA ? SPIB» SPICSPID.SPT) 106 IF( I.EQ.l.OR.I .EQ.NYP) GO TO 270 107 GO TO( 180,181,182,183) NCY 108 183 SPYD=SPYD+V( IVY)*SPID 109 182 SPYC=SPYC+V( IVY)*SPIC 110 181 SPYB=SPYB+V( 1VY)*SPIB 111 180 SPYA=SPYA+V( IVY)*SPIA 112 SPYT=SPYT+V( IVY)*SPT 113 TEK=TEK+EH( IVY)*SPT SU M ABSOLUTE VALUE OF ERROR EACH 6 SUBINTERVALS 114 IVY=IVY+1 115 IF(IVY.GE.7) IVY=1 116 IF( IVY.EQ.2 )GO TO 269 117 GO TO 280 118 269 EK=EK+DABS(TEK) •EK=ERROR IN Y INTEGRAL DUE TO WEDDLE RULE 119 TEK=SPT 120 GO TO 280 121 270 GO T01271, 272,273, 274) NCYiCOEFF. OF FIRST AND LAST TERMS IS 1, 122 274 SPYD=SPYD+SPID 123 273 SPYC=SPYC+SPIC 124 272 SPYB=SPYB+SPIB 125 271 SPYA=SPYA+SPIA 126 SPYT=SPYT+SPT 127 EK=EK+SPT 128 280 CONTINUE 129

EKI=EKI+EK*DA/140.D0 ' Y NORMALIZATION, WEDDLE RULE 130 STI=.3D0*DA*SPYT+STI 'TRANSMISSION INTEGRAL 131 GO TO(290, 291,292, 293) NCY 132 293 SYD=SYD+0.3D0*DA*SPYD 133 292 SYC=SYC+0.3D0*DA*SPYC 134 291 SYB=SYB+0.3D0*DA*SPYB 135

10 290 SYA=SYA+0.3D0*DA*SPYA 136 281 CONTINUE END OF Y INTEGRAL 137 VNORM=2.D0/RD**2 VNORM=2/RD**2 AND UNIT NORMALIZES 138 STIS=STI*VNORM THE INTEGRALS 139 SAFIS=SYA*VNORM INTERACTION IN "A" MATERIAL 140 SBFIS=SYB*VNORM INTERACTION IN 'B' MATERIAL 141 SCFIS=SYC*VNORM INTERACTION IN >C" MATERIAL 142 SDFIS=SYD*VNORM INTERACTION IN «D» MATERIAL 143 IF(SIGAT.LT.1.D-4)G0 TO 282 144 SFIS=SAFIS*SIGAF/SIGAT FISSION PART OF «A" INTERACTION 145 GO TO 283 146 282 SFIS=0.D0 147 283 SNOS=SFIS*AK+STIS NEUTRON S OUT PER NEUTRON HITTING SPHERE 148 ACROS=3.1416D0*RD**2 SOURCE CROSS SECTION 149 EM=EKI*1.D2/STI EM=ESTI MATE OF PER CENT ERROR IN STI 150 FTSOU=STIS SINGLE STRIKE TRANSMISSION PROBABILITY 151 FINSOU=l.D0-FTSOU SINGLE STRIKE INTERACTION PROBABILITY 152 FMULSO=SFIS SINGLE STRIKE FISSION PROBABILITY 153 FABSOU=FINSOU-FMULSO SINGLE STRIKE ABSORPTION PROBABILITY 154 FOUTSO=FTSOU+FMULSO*AK NEUTRON S OUT PER NEUTRON STRIKING SOURCE 155 FNOUT=l.D0-FOUTSO RESULTA NT LOSS 156 FPT=STIS*ACROS UNIT FL UX TRANSMISSION 157 FPI=F.INSOU*ACROS UNIT FL UX INTERACTION 158 FPA=FABSOU*ACROS UNIT FL UX ABSORPTION 159 FPF=SFIS*AK*ACROS UNIT FL UX FISSION 160 FNLS=FPI-FPF UNIT FL UX. NET LOSS 161 FQT=F*FPT PERCENT Q OF TRANSMISSION 162 FQI=F*FPI PERCENT Q INTERACTION 163 FQA=F*FPA PERCENT Q ABSORPTION 164 FQF=F*FPF PERCENT Q FISSION GAIN 165 FQL=F*FNLS PERCENT Q NET LOSS 166 WRITE(6.9)IGRAND,DY 167 WRITE(6»5) FTS0U,FINSOU ,FABSOU,SAFIS,SBFIS,SCFIS,SDFIS,FMULSO, 168 1F0UTS0»FN0UT,ACR0S,EM 169 WRITE(6»7)ACROS,FPT.FPI ,FPA,FPF,FNLS 170 WRITE(6»8)F,FQT,FQI »FQA ,FQF,FQL 171 GO TO 100 172 173 SUBROUTINE G RAND (W, DUMA, DUMB, DUMC,DUMD,DMDA) 174 IGRAND=IGRAN D+l 'COUNT OF SUBROUTINE CALL 175 DUMA=0.D0 •CONTAINS MULTIPLICATION PART OF INTEGRAND 176 DUMB=0.D0 177 DUMC=0.D0 178 DUMD=0.D0 179 A=0.D0 180 8=0. DO 181 C=0.D0 182 D=0.D0 183 GO TO (370,3 60,350.340) .NCY 184 340 D=(RDY-RCY)* 2. DO •THE AMOUNT OF A.B.C, AND D MATERIAL 185 350 C=(RCY-RBY)* 2. DO 'TRAVERSED IS A,B,C, AND D, RESPECTIVELY. 186 360 B=(RBY-RAY)« 2. DO 187 370 A=RAY*2.D0 188 DMD=-(SIGAT* A+SIGB*B+SI GC*C+SIGD*D) 189 DMDEX=DEXP(D MD) 190 DMDA=W*DMDEX •TRANSMISSION PART OF INTEGRAND 191 DEXD=DEXP(-S IGD*D/ 2. DO) 192 DEXC=DEXP(-S IGC*C/ 2. DO) 193 DEXB=DEXP(-S IGB*B/ 2. DO) 194 DEXA=DEXP(-S IGAT*A /2.D0 195

GO T0(400,40 1,402, 403) , NCY 196

403 DUMDA=W*(1-D MDEX) • IN TERACTION IN A,B,C,AND D MATERIAL 197 402 DUMCA=W*(DEX D-DMDE X/DEX D) •INTERACTION IN A,B, AND C MATERIAL 198 401 DUMBA=W*(DEX D*DEXC -DMDE X/( DEXD*DEXC) )• INTERACTION IN A AND B MATERIAL 199

400 DUMA=W*(DEXD *DEXC* DEXB- DMD EX/(DEXD*DEXC*DEXB) ) UNTERAC* IN A MAT. 200

GO T0(410,41 1,412, 413) , NCY 201

413 DUMD=DUMDA-D UMCA • IN TERACTION IN D MATERIAL 202 412 DUMC=DUMCA-D UMBA • IN TERACTION IN C MATERIAL 203 411 DUMB=DUMBA-D UMA • IN TERACTION IN B MATERIAL 204 410 RETURN 205 END 206

11 ) )

4.1.2 LIST OF THE INPUT DATA DECK

TITLE CARD. FORMAT t 13A6 »A2 )

EQUIVALENT SPHERICAL SOURCE FOR CYLINDRICAL SOURCE M-621 CARD 1

MACROSCOPIC CROSS SECTIONS AND INTEGRATION PARAMETERS. FORMAT ( 7G8 .4. 1 4 2 3456 789012 345678901234 567890123456789012 3456789012 3456789012345678901234567890 .865 4.305 1.16 .2811 .0 2.8 .122 100 CARD 2 SIGAA SIGAF SIGB SIGC SIGD AK F NY

I I I I I 1/6 NUMBER OF INTEGRA-

I I I I I TION STEPS

I I I I FLUX IN PERCENT OF SOURCE

I I I I STRENGTH PER CM**2

I I I NUMBER OF NEUTRONS PER FISSION I I ABSORPTION CROSS SECTION PER CM FOR D MATERIAL I ABSORPTION CROSS SECTION PER CM FOR C MATERIAL ABSORPTION CROSS SECTION PER CM FOR B MATERIAL FISSION CROSS SECTION PER CM FOR A MATERIAL ABSORPTION CROSS SECTION PER CM FOR A MATERIAL

SOURCE DIMENSIONS IN INCHES. FORMAT t 12G6 . 3 12 3456 78 9012 345678901234 5678901234567890123456 7 8 9012 34567 89012 34567 8901234 567890 1.913 .11 .071 CARD 3 DIA TBI TCI TDI

I I I D SHELL THICKNESS IN INCHES

I I C SHELL THICKNESS IN INCHES

I B SHELL THICKNESS IN INCHES OUTSIDE DIAMETER OF SOURCE IN INCHES

4.1.3 SIMPLIFIED SAMPLE OUTPUT

EQUIVALENT SPHERICAL S OURCE FOR THE CYLINDRICAL PU-BE M-62 1 SOURCE SOURCE DIMENSION INPUT IN INCHES, OD=1.913» B SHELL=0.110» C SHELL=0.071. MACROSCOPIC CROSS SECT IONS PER CM SIGAA= 1.865 SIGAF= 4.305 » SIGB= 1.16 SIGC= 0.2811 NEUTRONS PER FISSION = 2.80 NY=100. FOR A SINGLE NEUTRON S TRIKING THE SOURCE... THE TRANSMISSION PROBA BILITY IS .13567724+0 THE INTERACTION PROBAB ILITY IS .86432276+0 THE ABSORPTION PROBABI LITY IS .59782176+0 THE 'A' MATERIAL INTER ACTION PROBABILITY IS .38195382+0 THE

EQUIVALENT SPHERICAL SOURCE FOR THE CYLINDRICAL PU-BE A SOURCE SOURCE DIMENSION INPUT IN INCHES, OD=1.180» B SHELL=0.044. MACROSCOPIC CROSS SECTIONS PER CM

SIGAA= 1.214 , SIGAF= 2.802 . SIGB= 0.4106 NEUTRONS PER FISSION = 2.80 NY=100. FOR A SINGLE NEUTRON STRIKING THE SOURCE... THE TRANSMISSION PROBABILITY IS .26256084+0 THE INTERACTION PROBABILITY IS .73743916+0 THE ABSORPTION PROBABILITY IS .41973395+0 THE 'A' MATERIAL INTERACTION PROBABILITY IS .45535478+0 THE 'B' MATERIAL INTERACTION PROBABILITY IS .28208437+0 THE PROBABILITY FOR A FISSION ABSORPTION IS .31770520+0 NEUTRONS OUT PER NEUTRON STRIKE IS .11521354+1 NEUTRONS LOST PER NEUTRON STRIKE IS -.15213542+0 THE CROSS SECTION OF THE SOURCE IN CM**2 IS .70554114+1

12 . 6

THE PERCENT ERROR IN THE TRANSMISSION PROBABILITY IS .0185 IF THE MEASURED THERMAL FLUX IN PERCENT OF Q PER CM**2 IS .122 THE TRANSMISSION IN PERCENT OF Q IS .22600192+0 THE INTERACTION IN PERCENT OF Q IS .63475827+0 THE ABSORPTION IN PERCENT OF Q IS .36129028+0 THE r ISSION IN PERCENT OF Q IS .76571038+0 AND THE RESULTANT LOSS IN PERCENT OF Q IS -.13095211+0

EQUIVALENT SPHERICAL SOURCE FOR THE CYLINDRICAL AM-BE SOURCE SOURCE DIMENSION INPUT IN INCHES. OD=1.551. B SHELL=0.080» C SHELL=0.042. MACROSCOPIC CROSS SECTIONS PER CM

SIGAA= .1704 . SIGAF= 0.00082 . SIGB = 1.16 . SIGC= 0.281 NEUTRONS PER FISSION = 2.89 NY=100. FOR A SINGLE NEUTRON STRIKING THE SOURCE... THE TRANSMISSION PROBABILITY IS .34932104+0 THE INTERACTION PROBABILITY IS .65067896+0 THE ABSORPTION PROBABILITY IS .64994216+0 THE 'A' MATERIAL INTERACTION PROBABILITY IS ,15384751+0 THE 'B' MATERIAL INTERACTION PROBABILITY IS .41839021+0 THE «C« MATERIAL INTERACTION PROBABILITY IS .78441236^-1 THE PROBABILITY FOR A FISSION ABSORPTION IS .73680038-3 NEUTRONS OUT PER NEUTRON STRIKE IS .35145039+0 NEUTRONS LOST PER NEUTRON STRIKE IS .64854961+0 THE CROSS SECTION OF THE SOURCE IN CM#*2 IS .12189389+2 THE PERCENT ERROR IN THE TRANSMISSION PROBABILITY IS .0240 IF THE MEASURED THERMAL FLUX IN PERCENT OF Q PER CM**2 IS .121 THE TRANSMISSION IN PERCENT OF Q IS .51521920+0 THE INTERACTION IN PERCENT OF Q IS .95969683+0 THE ABSORPTION IN PERCENT OF Q IS .95861012+0 THE FISSION IN PERCENT OF Q IS .31406170-2 AND THE RESULTANT LOSS IN PERCENT OF Q IS .95655622+0

4.2 CYLINDRICAL CASE

4.2.1 LIST OF THE PROGRAM

IMPLICIT DOUBLE PREC I SI ON ( A-H ,0-Z ) 1 DIMENSION V(6) »T( 14) »EH<6> »AW(8) .RZZ(5) »KA(4) 2

) ALIM(RZABCD.DLRDP) =RZABCD/CX+ < DLRDP-RZABCD*S/CX *S 3

DATA V/2.D0.5.D0.1.D0.6.D0.1.D0.5.D0/ » EH/ 1 .DO .-6 . DO 4 115.D0.-20.D0.15.D0.-6.D0/.CC/1.D-8/ 5 1 F0RMAT(13A6,A2/7G8.4,2I3,G8.4) 6 3 FORMATI12G6.3) 7

4 -0RMAT(1H1,13A6,A2/' SOURCE D IMENS I ONS • 25X • / MACROSCOPIC CROSS SEC 8 1 '17X'/ INTEGRATION PARAMETERS'/' RA ='G14.6,< TRB ='G14.6»' 9

1 / SIGAA='G14.6» • AK ='G14.6»' / NZ ='16/' RB ='G14.6»' TRC 10

1 ='G14.6»' / SIGAF='G14.6» • F ='G14.6»» / NPHI ='16/' RC ='G1 11 14. 6»' TRD ='G14.6»' / SIGB ='G14.6»21X< / ET ='G14.6/' RD = 'G 12 114.6.' TLB ='G14.6»' / SIGC =«G14.6,21X' / NCY .='16/' DLA ='G14 13 1.6.' TLC ='G14.6»' / SIGD ='G14.6»21X' /• /21X' TLD = 14 l'G14.6.' /•/• IN INCHES. OUTSIDE LENGTH ='G12.6.' / COVER / WALL 15

1 THICKNESS / LEFT END / RIGHT END ' / 12X ' OUTS I DE D I AMETER= ' G 12 . 6 . 16 / 1' / B'4XG12.6.4X'/'G12.6. ' / • Gl 2. 6/42X • / C • 4XG1 2 .6 . 4X ' ' G12 . 17

1." /'G12.6/42X'/ D'4XG12.6,4X' /'G12.6. ' /'G12.6) 18 5 FORMAT13X/ 46H FOR A SINGLE NEUTRON STRIKING THE SOURCE.../ 19 120X35HTHE TRANSMISSION PROBABILITY IS 22XG16.8/ 20 120X35HTHE INTERACTION PROBABILITY IS 22XG16.8/ 21 120X35HTHE ABSORPTION PROBABILITY IS 22XG16.8/ 22 120X55HTHE 'A' MATERIAL ABSORPTION INTERACTION PROBABILITY IS 2XG16 23 1.8/20X52HTHE 'A' MATERIAL FISSION INTERACTION PROBABILITY IS 5XG16 24 1.8/20X44HTHE 'A' MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 25 120X49HTHE CLADDING MATERIAL INTERACTION PROBABILITY IS 8XG16.8 26

1 /16X' NEUTRONS OUT PER NEUTRON STRIKE IS ' 22XG16.8 27 1/20X'NEUTRONS LOST PER NEUTRON STRIKE '22XG16.8,' .' 28 1//46H THE AVERAGE CROSS SECTION OF THE SOURCE IS G13.8.' CM**2.' 29 1) 30

13 ' »

6 F0RMAT13X/ 62H FOR A SINGLE THERMAL NEUTRON STRIKING A SPHERE 31 10F RADIUS R=G13.8»' CM, WHICH JUST SURROUNDS THE SOURCE...'/ 32 120X40HTHE PROBABILITY OF MISSING THE SOURCE IS17XG16.8/ 33 120X'THE TRANSMISSION (INCLUDING MISS) PROBABILITY IS'9XG16.8/ 34 120X35HTHE INTERACTION PROBABILITY IS 22XG16.8/ 35 120X35HTHE ABSORPTION PROBABILITY IS 22XG16.8/ 36 120X44HTHE 'A' MATERIAL INTERACTION PROBABILITY IS 13XG16.8 / 37 120X57HTHE PROBABILITY FOR A FISSION ABSORPTION IS G16 38

1.8/16X' NEUTRONS OUT PER NEUTRON INTO SPHERE ' 2 1XG1 6. 8 » 2H ./3XiT 39 1HE ERROR IN THE TRANSMISSION PROBABILITY IS ESTIMATED TO BE 'G12.3 40

• l»i PER CENT. ) 41

7 FORMAT13X/ 44H FOR A FLUX OF ONE NEUTRON PER CM**2-SEC . ( G12 . 5 42 151H NEUTRONS PER SEC) STRIKING THE THE ABOVE SPHERE.../ 43 120X26HTHE TRANSMISSION IS G16.8.20H NEUTRONS PER SECOND/ 44 120X26HTHE INTERACTION IS G16.8.20H NEUTRONS PER SECOND/ 45 120X26HTHE ABSORPTION IS G16.8»20H NEUTRONS PER SECOND/ 46 120X26HTHE FISSION EMISSION IS G16.8»20H NEUTRONS PER SECOND/ 47 116X26HAND THE RESULTANT LOSS IS 4XG16.8,' NEUTRONS PER SECOND.') 48 8 FORMATI3X/ 35H IF THE MEASURED THERMAL FLUX IS G12.4, ' PER CE 49 INT OF PER CM**2 THEN...'/ 50 120X26HTHE TRANSMISSION IS G16.8,14H PER CENT OF Q/ 51 120X26HTHE INTERACTION IS G16.8»14H PER CENT OF Q/ 52 120X26HTHE ABSORPTION IS G16.8»14H PER CENT OF Q/ 53 120X26HTHE FISSION EMISSION IS G16.8»14H PER CENT OF Q/ 54 116X26HAND THE RESULTANT LOSS IS 4XG16.8,' PER CENT OF Q.') 55 9 FORMAT! lX'THE NUMBER OF GRAND CALLS WAS'I8»'» 2ND 3RD, AND 4TH ORD 56 , 1ER COUNTS.KB, KC, AND KD WERE • I 8 • , ' I 8 , • , AND ' I 3 , / IX ' THE LARGEST R 57 1ELATIVE ERRORS, ERL1, ERL2, AND ERL3, ENCOUNTERED AFTER THE 1ST, 2 58 X 1ND» AND 3RD SUBDIVISIONS, THE LARGEST • / 1 ' SUBDI V I S I ON , DZ , ALONG 59

] THE Z-AXIS, AND THE LARGEST 3RD ORDER SUBDIVISION, DWD. OF THE W- 60 1AXIS, WERE'/ 5<4X,G16.8)> 61 100 KB=0 'SECOND ORDER COUNT, STATEMENT 60+2, SUB IN 62 KC=0 'THIRD ORDER COUNT, STATEMENT 65+2 SUB IN 63 KD=0 'FOURTH ORDER COUNT, STATEMENT 121 SUB IN 64 READ (5,1 )T,SIGAA,SIGAF,SIGB,SIGC,SIGD,AK,F,NPHI ,NZ,ET 65

IF(ET.LT..1D-2)ET=.1D-1 ET=SPEC I F IED RELATIVE ERROR IN SUBROUTINE IN 66 READ (5 ,3) AA,AB,TRBI »TLBI »TSB»TRCI ,TLCI ,TSC,TRDI «TLDI ,TSD 67 IGRAND=0 'COUNTER RESET FOR NUMBER OF SUB GRAND CALLS 68 NCY=4 'AA=OUTSIDE LENGTH IN INCHES 69 IFtTSD. LT.CC.AND.TRDI.LT. CC.AND.TLDI.LT. CO NCY=3 ' 3 LAYERS 70 IFtTSC.LT.CC.AND.TRCI.LT.CC.AND.TLCI.LT.ee) NCY=2 ' 2 LAYERS 71

IF( TSB.LT.CC.AND.TRBI.LT.CC.AND.TLBI.LT.ee) NCY=1 ' 1 LAYER 72 TRB=2.54D0*TRBI THE INPUT DIMENSIONS WERE IN INCHES 73 TLB=2.54D0*TLBI F=THE NEUTRON FLUX IN PERCENT OF Q 74 TRC=2.54D0*TRCI AK=NEUTRONS PER FISSION 75 TLC=2.54D0*TLCI NCY=THE NUMBER OF DIFFERENT MATERIALS 76 TRD=2.54D0*TRDI T=TITLE 77 TLD=2.54D0*TLDI TRB= THICKNESS, RIGHT END, B CYLINDER 78 RD=AB*2.54D0/2.D0 OUTSIDE RADIUS 79 RC=RD-2.54D0*TSD FOR ONE LAYER A,B,C, AND D DIMENS. ARE EQUAL 80 RB=RC-2.54D0*TSC FOR TWO LAYERS B,C, AND D DIMENSIONS ARE EQU AL 81 RA=RB-2.54D0*TSB FOR THREE LAYERS C AND D DIMENSIONS ARE EQUA 82 DLA=2.54D0*AA-TRB-TRC-TRD-TLB-TLC-TLD 83 BLA=DLA/2.D0 ONE-HALF OF INSIDE LENGTH 84 DRB=BLA+TRB AB=OUTSIDE DIAMETER OF SOURCE IN INCHES 85 DRC=DRB+TRC SIGAA=MACROSCOPIC ABSORBTION CROSS SECTION OF 86

DRD=DRC+TRD MATERIAL A IN 'PER CM. < . 87 DLB=BLA+TLB SIGAF= FISSION IN 'A' IN PER CM. 88 DLC=DLB+TLC SIGB=ABSORPTION IN 'B' 89 DLD=DLC+TLD TLD=THICKNESS, LEFT END, CYLINDER D 90 PTR=DSQRT(DRD**2+RD**2) RA=RADIUS OF CYLINDER A 91 RTL=DSQRT(DLD**2+RD**2) 92 R = RTL 93 IF(RTR.GT.RTL) R=RTR R=SURROUNDING SPHERE RADIUS 94 ELM=R«0.001D0 SEE ST 210+4. SKIPS W INTEGR.IF LIM.TOO SMALl 95 ERL1=O.DO LARGEST ERROR IN THE 1ST, 2ND, AND 3RD 96 ERL2=0.D0 ORDER WEDDLE DIVISIONS IN SUBROUTINE IN. 97 ERL3=0.D0 98 DWD=0.D0 LARGEST DIVISION OF W-AXIS, SUBROUTINE IN 99 DZ=0.D0 LARGEST DIV. OF Z-AXIS IS MAX VALUE OF DA 100 SIGAT=SIGAF+SIGAA SIGAT=TOTAL CROSS SECTION, 'A' MATERIAL 101 RK=DLD NZ=NO. OF DIVS. OF Z FOR WEDDLE INTEGRATION 102

14 )

IF(DRD.GT.DLD)RK=DRD 'RK=LARGER OF DRD AND DLD 103

U=DATAN2(RK,RD) • ARCTAN < RK/RD 104 AG=U*(R**2) 105 ARE=RD*RK 106 AD=AG-ARE 'AD = CROSS SECTIONAL AREA OF SPHERE ABOVE 107 NPHIP=6*NPHI+1 'CYLINDER AND NOT IN THE Z INTEGRAL RANGE. 108 VKB=6.D0*NPHI 'VKB = THE NUMBER OF PHI INTERVALS PER 90 DEGR 109 DB=1.5707963D0/VKB 'DB= PHI INTERVAL IN RADIANS 110

WRITE(6»4)T,RA,TRB,SIGAA,AK,NZ,RB,TRC,SIGAF»F,NPHI ,RC , TRD , S I GB , ET 111 1 »RD»TLB,SIGC,NCY,DLA,TLC,SIGD,TLD,AA,AB,TSB»TLBI »TRBI »TSC , TLCI 112 ] TRCI ,TSD,TLDI »TRDI 113 RZZ(1)=0.D0 114 RZZ(2)=RA 115 RZZ(3)=RB 116 RZZ(4)=RC 117 RZZ(5)=RD 118 EJI=0.D0 119 EKI=0.D0 120 STI=0.D0 121 SFI=0.D0 122 SMI=0.D0 123 ANZ=NZ/RD 'NZ=APPROXIMATE NO. OF Z INTERVALS 124 KA< 1>=RA*ANZ+.5D0 'THIS IS TO SUBDIVIDE THE Z-LAYERS IN 125 KA(2)=(RB-RA)*ANZ+.5D0 'PROPORTION TO THE THICKNESS OF EACH LAYER 126 KA(3)=(RC-RB)*ANZ+.5DO 'FROM ZERO TO RA , FROM RA TO RB . FROM RB 127 KA(4) = (RD-RC)*ANZ+.5D0 'TO RC, AND FROM RC TO RD. Z-I NTEGRAT I ON WILL 128 DO 281 IAR=1.NCY 'ALWAYS OCCUR AT LAYER BOUNDARIES AND AVOID 129 IF(KA( IAR) ,LT.1)KA< IAR)=1 'DISCONTINUITIES. 130 IF( (RZZ( IAR+1)-RZZ( IAR) ) .LT.ELM)GO TO 281'IN CASE OF ZERO SIDEWALL 131

DA=(RZZ( IAR+1)-RZZ( IAR) )/(6.D0*KA( IAR) ) 132 IF(DA.GT.DZ)DZ=DA 'LARGEST DIVISION OF Z-AXIS 133 NZP=6*KA( IAR1+1 'THE NUMBER OF Z STEPS 134 IVZ=2 'IVZ= Z INTEGRATION WEDDLE RULE COEF INDICATOR 135 SPZT=0.D0 'Z PARTIAL SUM RESET OF TRANSMISSION, 136 SPZM=0.D0 'MULTIPLICATION, AND 137 SPZMIS=0.D0 'MISSES 138 EJ=0.D0 'ERROR RESET FOR Z INTEGRAND IN DO LOOP 280 139 EK=0.D0 'ERROR RESET FOR Z INTEGRAL IN DO LOOP 280 140 TEK=0.D0 'TEMPORARY EK 141 DO 280 1=1, NZP 'BEGIN. Z INTEGR.FROM O-RA-RB-RC-RD 142 IVP=2 'ANGLE INTEGRATION WEDDLE RULE COEFFICIENT INDICATOR. 143 SPPT = 0.D0 'ANGLE PARTIAL SUM RE-SET OF TRANSMISSION, 144 SPPM=0.D0 'MULTIPLICATION, AND 145 SPPMIS=0.D0 'MISSES 146 EF=0.D0 'ERROR RESET FOR PHI INTEGRAND IN DO LOOP 250 147 EG=0.D0 'ERROR RESET FOR PHI INTEGRAL IN DO LOOP 250 148 TEG=0.D0 'TEMPORARY EG 149

Z=DA*DBLE ( 1-1 )+RZZ( IAR) 150 RZ=DSQRT(R**2-Z**2) 151 RZD=0.D0 152 RZC=0.D0 153 RZB=0.D0 154 RZA=0.D0 155

GO TO ( 173,172,171 ,170) ,NCY 156 170 IF(RD.GT.Z)RZD=DSQRT(RD**2-Z**2) 157 171 IF(RC.GT.Z) RZC=DSQRT(RC**2-Z**2) 'HIT C LAYER, IF THERE IS ONE 158

172 IF(RB.GT.Z) RZB=DSQRT < RB**2~Z**2 ) 'HIT B LAYER, IF THERE IS ONE 159

173 IF(RA.GT.Z) RZA = DSQRT ( RA**2-Z**2 ) 'HIT A LAYER 160 DO 250 J=1,NPHIP 'BEGINNING OF PHI INTEGRATION FOR PHI FROM 161 PHI=DB*DBLE( J-l) '0 TO 90 DEGREES ONLY. PHI IS THE ANGLE 162 S=DSIN(PHI) 'BETWEEN THE Y-AXIS AND W-AXIS 163 CX=DCOS(PHI) 164 SPWT=0.D0 'W PARTIAL SUM RESET OF TRANSMISSION, 165 SPWM=0.D0 'AND MULTIPLICATION 166 IFLAG=3 'PHI NOT EQUAL TO OR 90 DEGREES 167 IF(J.EQ.l) GO TO 180 168 IF( J.EQ.NPHIP) GO TO 185 169 GO TO 200 170 180 IFLAG=1 'PHI=0 DEGREES 171 GO TO( 184,183,182,181) ,NCY 172 181 AW(1)=-RZD 'W(l) IS THE POSITION ALONG THE W-AXIS THAT A 173 AW(8)=RZD 'NEUTRON, INCIDENT PERPENDICULARLY FROM THE 174

15 )

182 AW(2)=-RZC 'RIGHT OR LEFT SIDE OF THE W-AXIS WILL JUST 175 AW<7)=RZC 'STRIKE THE LEFT OUTER CORNER OF THE D 176 183 AW(3)=-RZB 'MATERIAL. AW<2), AW(3) » AND AW<4) WILL JUST 177 AW(6)=RZB 'STRIKE THE C,B, AND A LAYERS* RESPECTIVELY. 178

184 AW(4)=-RZA 'AW(5)» AW(6)» AW ( 7 ) . AND AW(8) WILL JUST 179 AW(5)=RZA 'STRIKE THE UPPER RIGHT OUTER CORNER OF THE 180 GO TO 210 'A. B, C. AND D MATERIALS, RESPECTIVELY. THIS181 185 IFLAG=2'PHI=90 DEGREES. 'AVOIDS DISCONTINUITIES IN THE DERIVATIVES FOR182 GO TO( 189.188, 187*186) .NCY'THE WEDDLE RULE. 183 186 AW( 1)=-DLD 184 AW(8)=DRD 185 187 AW(2)=-DLC 186 AW(7)=DRC 187 188 AW(3)=-DLB 188 AW(6)=DRB 189 189 AW(4)=-BLA 190 AW(5)=BLA 191 GO TO 210 192 200 GO TO(204,203,202,201) ,NCY 193 201 AW( 1)=-ALIM(RZD»DLD) 194 AW(8)=ALIM(RZD.DRD) 195 202 AW(2)=-ALIM(RZC.DLC) 196 AW(7)=ALIM(RZC»DRC) 197 203 AW(3)=-ALIM(RZB,DLB) 198 AW(6)=ALIM(RZB»DRB) 199 204 AW(4)=-ALIM(RZA,BLA) 200 AW(5)=-AW<4) 201 210 LL=5-NCY 202 LU=3+NCY 203 DELW=AW(LU+1 )-AW(LL) 'RQ=2* THE PORTION OF THE SPHERE CUT BEYOND 204 RQ=2.D0*(2.D0*RZ-DELW) 'CYLINDER BECAUSE OF INCIDENCE FROM TWO 205 IFIDELW.LT. ELM)GO TO 231 'DIRECTIONS 206 EZ=0.D0 207 DO 230 I2=LL»LU 208 CALL IN( AW( 12) ,AW( 12+1) ,E,SPT,SPM) 209 EZ=EZ+E 210 SPWT=SPWT+SPT 211 SPWM=SPWM+SPM 212 230 CONTINUE 'END OF W INTEGRAL DO LOOP, WHICH BECOMES 213 231 GO T0(240, 240, 235) ,IFLAG' INTEGRAND FOR PHI INTGR. WITH ABS. ERROR EZ 214 235 SPPT=SPPT+V( IVP)*SPWT 215 SPPM=SPPM+V( IVP)*SPWM 216 SPPMIS=SPPMIS+V( IVP)*RQ 'MISS CYLINDER CALCULATION 217 EF=EF+(V( IVP)* EZ)**2" 'ERROR**2 IN PHI IN. DUE TO INTEGRND. UNCRT. 218

) TEG=TEG+EH( I VP *SPWT ' SUM ABSOLUTE VALUE OF ERROR EACH 6 SUBDIVISIONS 219 IVP=IVP+1 »EH( IVP)=WEDDLE RULE ERROR CONSTANTS 220 IFUVP.GE.7) IVP=1 221 IF( IVP.EQ.2JGO TO 239 222 GO TO 250 223 239 EG=EG+DABS(TEG) 'EG=ERROR IN PHI INTEGRAL DUE TO WEDDLE RULE 224 TEG=SPWT 225 GO TO 250 226 240 SPPT=SPPT+SPWT 'COEFF. FOR FIRST AND LAST TERMS IS ONE 227 SPPM=SPPM+SPWM 228 SPPMIS=SPPMIS+RQ 229 EG=EG+SPWT 230 EF=EF+EZ**2 231 250 CONTINUE 'END OF PHI INTEGRAL DO LOOP, WHICH BECOMES 232 SPPT=SPPT*0.3D0*DB 'INTEGRAND FOR THE Z INTEGRAL 233 SPPM=SPPM*0.3D0*DB 'THIS IS THE WEDDLE RULE NORMALIZATION 234 SPPMIS=SPPMIS*0.3D0*DB 'FOR THE PHI PART 235 EF=.3D0*DB*DSQRT(EF> 236 EG=EG*DB/140.D0 237 EI=DSQRT

) TEK=TEK+EH( I VZ *SPPT ' SUM ABSOLUTE VALUE OF ERROR EACH 6 SUBDIVISIONS 244 IVZ=IVZ+1 245 IFUVZ.GE.7) IVZ=1 246

16 IF( IVZ.EQ.2)G0 TO 269 247 GO TO 280 248 269 EK=EK+DABS(TEK> 'EK=ERROR IN Z INTEGRAL DUE TO WEDDLE RULE 249 TEK=SPPT 250 GO TO 280 251 270 SPZT=SPZT+SPPT •COEFF. OF FIRST AND LAST TERMS IS 1, 252 SPZM=SPZM+SPPM 253 SPZMIS=SPZMIS+SPPMIS 254 EJ=EJ+EI**2 255 EK=EK+SPPT 256 280 CONTINUE 257 EJI=EJI+.3D0*DA*DSQRT(EJ) Z NORMALIZATION. WEDDLE RULE 258 EKI=EKI+EK*DA/140.D0 259 STI=.3D0*DA*SPZT+STI •TRANSMISSION INTEGRAL 260 SFI=.3D0*DA*SPZM+SFI 'FISSION INTEGRAL 261 SMI=.3D0*DA#SPZMIS+SMI 'MISSES INTEGRAL 262 281 CONTINUE 'END OF Z INTEGRAL 263 AR =3.1416D0*(R#*2) 'SPHERE CROSS SECTION 264 VNORM=2.DO/(3.1416D0*AR) 'UNIT NORMALIZES THE INTEGRALS 265 EL=DSQRT(EJI**2+EKI**2) 266 STIS=STI*VNORM 267 SAFIS=SFI*VNORM 'INTERACTION IN 'A' MATERIAL 268 IF(SIGAT.LT.1.D-4)G0 TO 282 269 SFIS=SAFIS*SIGAF/SIGAT 'FISSION PART OF 'A' INTERACTION 270 GO TO 283 271 282 SFIS=0.D0 272 283 SMIS=SMI*VNORM 273 SLM=2.D0*AD/AR 'UNIT NORMALIZED. LUMP MISS PART 274 SNOS=SFIS*AK+SLM+SMIS+STIS 'NEUTRONS OUT PER NEUTRON HITTING SPHERE 275 FMIS=SLM+SMIS SINGLE NEUTRON MISS PROBABILITY 276 ACR0S=AR*(1.D0-FMIS) AVERAGE SOURCE CROSS SECTION 277 FTRANS=SLM+STIS+SMIS SINGLE NEUTRON TRANSMISSION PROBABILITY 278 EM= EL* l.D2*VNORM/F TRANS EM=ESTIMATE OF PER CENT ERROR IN FTRANS 279 FINT=1.D0-FTRANS SINGLE NEUTRON INTERACTION PROBABILITY 280 FMUL=SFIS SINGLE NEUTRON FISSION PROBABILITY 281 FABS=FINT-FMUL SINGLE NEUTRON ABSORPTION PROBABILITY 282 FNO=FMUL*AK+FTRANS NEUTRONS OUT PER NEUTRON STRIKING SPHERE 283 FTSOU=(FTRANS-FMIS)*AR/ACROS 'SINGLE STRIKE TRANSMISSION PROBABILITY 284 FINSOU=l.D0-FTSOU SINGLE STRIKE INTERACTION PROBABILITY 285 FSAFIS=SAFIS/(1.D0-FMIS 'SINGLE STRIKE 'A' INTERACTION PROBABILITY 286

FMULSO=FMUL/ < 1.D0-FMIS) SINGLE STRIKE FISSION PROBABILITY 287 AAIP=FSAFIS-FMULSO SINGLE STRIKE 'A' ABSORPTION INTERACT. PROBAB.288 BCDIP=FINSOU-FSAFIS SINGLE STRIKE CLADDING INTERACTION PROBABIL. 289 FABSOU=FINSOU-FMULSO SINGLE STRIKE ABSORPTION PROBABILITY 290 FOUTSO=FTSOU+FMULSO*AK NEUTRONS OUT PER NEUTRON STRIKING SOURCE 291 FNOUT=l.D0-FOUTSO RESULTANT LOSS 292 FPT=FTRANS*AR UNIT FLUX TRANSMISSION 293 FPI=FINT*AR UNIT FLUX INTERACTION 294 FPA=FABS*AR 'UNIT FLUX ABSORPTION 295 FPF=FMUL*AR*AK UNIT FLUX FISSION 296 FNLS=FPI-FPF 'UNIT FLUX. NET LOSS 297 FQT=F*FPT PERCENT OF TRANSMISSION 298 FQI=F*FPI 'PERCENT INTERACTION 299 FQA=F*FPA 'PERCENT ABSORPTION 300 FQF=F*FPF PERCENT FISSION GAIN 301 FQL=F*FNLS 'PERCENT NET LOSS 302 WRITE (6 .9) IGRAND,KB.KC,KD.ERL1,ERL2.ERL3.DZ.DWD 303 WRITE (6. 5) FTSOU.FINSOU.FABSOU.AAIP.FMULSO.FSAFIS.BCDIP.FOUTSO. 304 1FNOUT.ACROS 305 WRITE(6.6)R,FMIS»FTRANS»FINT.FABS.SAFIS.FMUL.FNO.EM 306 WRITE(6.7)AR.FPT,FPI .FPA .FPF.FNLS 307 WRITE(6.8)F»FQT»FQI .FQA.FQF.FQL 308 GO TO 100 309 310

SUBROUTINE I N ( FLL , FUL »E . ANS.QE ) 'TO GIVE THE WEDDLE RULE- INTEGRAL TO 311 DW1=(FUL-FLL)/36.D0 'A SPECIFIED RELATIVE ERROR ET. 312 DW2=DW1/6.D0 'IF THE SUBDIVISION ALONG THE W-AXIS IS 6 313 IF(DW2.GT.DA)GO TO 65 'TIMES LARGER THAN IN THE Z-DIRECTION, THEN 314 IF(DW1.GT.DA)G0 TO 60 'IT IS AUTOMATICALLY DIVIDED DOWN BY 6 OR 36 315 CALL PCE(FLL.FUL.ANS.AE.QE) 316 IF(ANS.LT.1.D-4)G0 TO 50 317 EE=DABS< AE/ANS) 318

17 )

IF(EE.GT.ERL1)ERL1=EE 'ERL1 IS LARGEST ERROR IN 1ST ORDER WEDDLE DIV.319 IF(EE.GT.ET)GO TO 60 'FLL=LOWER LIMIT 320 E=AE «FUL=UPPER LIMIT 321

RETURN • ANS=INTEGRAL 322 50 E=0.D0 323 RETURN 324 60 ADIV=6.D0 325 IA1=6 326 KB=KB+1 'RESET COUNTER IN STATEMENT 100 327 GO TO 70 328 65 ADIV=36.D0 329 IA1=36 330 KC=KC+1 '3RD ORDER COUNT 331 DW3=DW2/6.D0 332 IF(DW3.GT.DWD)DWD=DW3 'DWD IS LARGEST 3RD ORDER SUBDIV. OF W-AXIS 333

70 ED=(FUL-FLL) /ADIV ' PCE RETURNS ABSOLUTE ERROR. 334 P1I=0.D0 335 OE=0.D0 336 E=0.D0 337 FL2=FLL 338 FU2=FLL+ED 339 DO 120 IA=1»IA1 340 CALL PCE(FL2»FU2.PI tPEI »QF) "2ND ORDER OR 3RD ORDER 341 P1I=P1I+PI 342 QE=QE+QF 343 E=E+PEI 344 FL2=FU2 'INCREMENT LIMITS 345 120 FU2=FU2+ED 346 ANS=P1I 347 IF( ANS.LT.1.D-41G0 TO 122 348 EE=DABS( E/ANS) 349 IF( IA1.EQ.36JGO TO 121 350

IF(EE.GT.ERL2)ERL2=EE • ERL2 IS LARGEST ERROR IN 2ND ORDER WEDDLE DIV.351 IF(EE.GT.ET)GO TO 65 '4TH ORDER COUNT ,KD , USED HERE TO COUNT NO OF 352 RETURN 'TIMES 3RD ORDER WEDDLE ERROR FAILS TO 353 121 IF(EE.GT.ET)KD=KD+1 'SATISFY RELATIVE ERROR ET IN INPUT 354 IF(EE.GT.ERL3)ERL3=EE 'ERL3 IS LARGEST ERROR IN 3RD ORDER WEDDLE DIV.355 RETURN 356 122 E=0.DO 357 RETURN 358 359

) SUBROUTINE PCE ( FL , FU , AI AE ,QC » GI VES THE WEDDLE CORRECT SUM BETWEEN FL360 DIMENSION VI I (7) ,VIE(7) 'AND FU USING 6 SEGMENTS. 361 DATA VII/l.D0»5.D0,l.D0»6.D0»l.D0»5.D0»l.D0/.VIE/l.D0»-6.D0»15.D0f 362 1-2G.D0»15.D0,-6.D0,1.D0/,CA/3.D-1/, CB/ . 71428571D-2/ 363 EC=(FU-FL)/6.D0 'AI IS THE FIRST INTEGRAL. TRANSMISSION PART 364 PI=0.D0 'QC IS THE SECOND INTEGRAL, MULTIPLICATION PART365 QB=0.D0 'AE IS THE ABSOLUTE ERROR IN THE FIRST INTEGRAL366 PE=0.D0 'EC IS THE INCREMENT SIZE 367 X=FL 'X IS THE INTEGRAND EVALUATION POINT 368 DO 100 ID=1,7 369 CALL GRAND(X,3A,QG) 'QA IS THE MULTIPLICATION PART. INTEGRAND 370 PI=PI+QG*VII (ID) 'QG IS THE TRANSMISSION PART, INTEGRAND 371

QB=QB+VI I ( ID)*QA 'PI IS THE TRANSMISSION PART PARTIAL SUM 372 PE=PE+QG*VIE< ID) 'QB IS THE MULTIPLICATION PARTIAL SUM 373 100 X=X+EC 374 AI=CA#EC*PI 'PE IS THE ERROR PARTIAL SUM (6TH DIFFERENCE) 375 AE=DABS(CB*EC*PE) 'QC IS THE MULTIPLICATION PART ANSWER 376 QC=CA*EC*QB 377 RETURN 378 379 SUBROUTINE GRANDt W.DUMC »DUMD 380 C FINAL EVALUATION OF THE INTEGRAND, WHICH IS PUT INTO PCE 381 C CONTRIBUTIONS ARE ADDED FOR NEUTRONS INCIDENT ON SOURCE FROM OPPOSITE 382 C DIRECTIONS, SO THAT THE PHI INTEGRAL IS CARRIED OUT FROM TO PI/2 383

C INSTEAD OF FROM TO PI. DUMD = TRANSM I SS I ON PART OF INTEGRAND. 384 IGRAND=IGRAND+1 385 DUMC=0.D0 'CONTAINS MULTIPLICATION PART OF INTEGRAND 386 A=0.D0 387 B=0.D0 388 C=0.D0 389 D=0.D0 390

18 ( ) > )

GO TO(90, 60, 30,10) , NCY 'NCY=NUMBER OF LAYERS 391 10 CALL C2( IB,DRD»DLD,RZD,W,XID,YID,D> 392 GO TO (20,30) , IB 393 20 DUMD=2.D0 "TRANSMISSION. IT MISSES IN BOTH DIRECTIONS 394 RETURN 395 30 CALL C2< IB,DRC»DLC,RZC»W,XIC»YIC»C) 396 IFt IB.EQ.2) GO TO 60 397 IF(NCY.EQ.3) GO TO 20 398 GO TO 130 399 60 CALL C2 IB,DRB,DLB,RZB,W,XIB,YIB,B) 400 IF( IB.EQ.2) GO TO 90 401 IFtNCY.EQ.2) GO TO 20 402 GO TO 130 403 90 CALL C2( IB,BLA»BLA»RZA»W,XIA,YIA,A) 404 IFIIB.EQ.2) GO TO 130 405 IF(NCY.EQ.l) GO TO 20 406

130 GO TO ( 170,160.150,140) .NCY 407 140 D=D-C 'THE AMOUNT OF A,B,C» AND D MATERIAL 408 150 C=C-B 'TRAVERSED IS A,B,C, AND D, RESPECTIVELY. 409 160 B=B-A 410 170 DUM=-(SIGAT*A+SIGB*B+SIGC*C+SIGD*D) 411 DUMD=2.D0*DEXP(DUM) 'TRANSMISSION 412 IF(IB.EQ.l) RETURN 'DID NOT HIT A MATERIAL. NO FISSION 413 DIDI=0.D0 414 DIDO=0.D0 . 415 DICI=0.D0 416 DICO=0.D0 417 DBPBA=-SIGAT*A 418 DBPBE=(1.D0-DEXP(DBPBA) 419 GO TO (220,210,200,190) , NCY 420

) ) 190 DIDI=DSQRT< ( X I D-X I C **2+ ( Y I D-Y I C **2 ) 'ENTRY THICK. FOR FISSION CALC. 421 DIDO=D-DIDI 'SAME, FROM OTHER SIDE 422

200 DICI=DSQRT( ( X I C-X I B > **2+ ( YIC-YI3)**2) 423 DICO=C-DICI 424

210 DIBI=DSQRT( ( X I B-X I A ) **2+ ( Y I B- Y I A ) **2 425 DIBO=B-DIBI 426

DUMBI=-(SIGD*DIDI+SIGC*DICI+SIGB*DIBI ) 427 DUMBO=-(SIGD*DIDO+SIGC*DICO+SIGB*DIBO) 42 8

DUMCI=DEXP(DUMBI ) 'FISSION EFFECT 429 DUMCO=DEXP(DUMBO) 'FROM OPPOSITE SIDE 430 DUMC=( DUMCI+DUMCO)*DBPBE 431 RETURN 432 220 DUMC=2.D0*DBPBE 433 RETURN 434 435

SUBROUTINE C2 ( I B, DR , DL , C2R , C2W , X I , Y I , DTHRU 436

YPF(X)=C2W*CX-S*(X-C2W*S )/CX • Y-COORD I NATE , GIVEN X, PHI, AND C2W(=W)437

XPF( Y)=C2W*S + CX*(C2W*CX-Y) /S ' X-COORD I NATE , GIVEN Y, PHI, AND C2W(=W)438 IF( IFLAG.EQ.DGO TO 101 439 IF( IFLAG.EQ.21GO TO 102 440 TR=YPF(DR) 441 l"L = YPF(-DL) 442 DTHRU=0.D0 443 IF1C2R.GT.TR) GO TO 30 444 20 IB=1 'NEUTRON MISSES CYLINDER. 445 RETURN 446 30 IF(-C2R.GE.TL) GO TO 20 447 IB=2 'NEUTRON HITS CYLINDER. 448 IFI-C2R.GT.TR) GO TO 60 449 XI=DR 'NEUTRON HITS END 450 YI=TR 451 GO TO 70 452 60 XI=XPF(-C2R) 'NEUTRON HITS BOTTOM. 453 YI=-C2R 454 70 IFIC2R.GT.TL) GO TO 90 455 XO=XPF(C2R) 'NEUTRON EXITS FROM THE TOP. 456 YO=C2R 457 GO TO 100 458 90 XO=-DL 'NEUTRON EXITS FROM THE END 459 YO=TL 460 100 DTHRU=DSQRT( (XI-XO)**2+( YI-YO)**2) 461 RETURN 462

19 ) )

101 DTHRU=DR+DL 463 XI=DR 464 YI=C2W 465 IB = 2 466 RETURN 467 102 DTHRU=2.D0*C2R 468 XI=C2W 469 YI=-C2R 470 IB = 2 471 RETURN 472 END 473

4»2.2 LIST OF THE INPUT DATA DECK

TITLE CARD. FORMAT ( 13A6 , A2 SOURCE M-621 PU-BE 80 GRAM, TA AND STAINLESS STEEL ENCAPSUL ATED CARD 1 MACROSCOPIC CROSS SECTIONS AND INTEGRATION PARAMETERS. FORMA T(7G8.4»2 I3.G8.4) 234 567 89012345678901234567890123456789012 34 567890123 45678901 234567890 1234567890 .865 4.305 1. 16 .2811 .0 2.{ .122 10 1 0.01 CARD 2 SIGAA SIGAF SIG3 SIGC SIGD AK F N- NZ ET

I PHI I I

I I I MINIMUM FRACTION-

I I AL ERROR ALLOWED

I I IN SUBRO UTINE PCE

I 1/ 6 NUMBER OF Z INTE-

I GR ATION STE PS

I 1/6 N UMBER OF PHI INTE-

I GRATI ON STEPS

I FLUX IN PERCE NT OF SOU RCE

I STRENGTH PER CM**2 NUMBER OF NEUTRONS PE R FISSION ABSORPTION CROSS SECTION PER CM FOR D MATERIAL ABSORPTION CROSS SECTION PER CM FOR C MATERIAL ABSORPTION CROSS SECTION PER CM FOR B MATERIA L FISSION CROSS SECTION PER CM FOR A MATERIAL ABSORPTION CROSS SECTION PER CM FOR A MATERIAL

SOURCE DIMENSIONS IN INCHES. FORMAT ( 12G6 . 3 12 34567890123456789012 34 56789012345678901234567890123 4567 89 012345678901234567890 2.72 1.310 .100 .250 .070 .250 .100 .03 .0 .0 .0 CA AA AB TRBI TLBI TSB TRCI TLCI TSC TRDI TLDI TSD

I I I I D SIDE WALL IN IN.

I I I D LEFT END IN INCHES

I I D RIGHT END IN INCHES

I C SIDE WALL IN INCHES C LEFT END IN INCHES

I I I I C RIGHT END IN INCHES

I I I B SIDE WALL IN INCHES

I I B LEFT END IN INCHES

I B RIGHT END IN INCHES OUTSIDE DIAMETER IN INCHES OUTSIDE LENGTH IN INCHES

4.2.3 SIMPLIFIED SAMPLE OUTPUT

CYLINDRICAL PU-BE M-621 SOURCE. ENCAPSULATED IN TANTALUM AND ST. STEEL, SOURCE DIMENSION INPUT IN INCHES, L = 2.72, OD = 1.31, B WALL = 0.07, B ENDS = 0.25 AND 0.1, C WALL = 0.03, C ENDS = 0.1 AND 0.25. MACROSCOPIC CROSS SECTIONS PER CM

SIGAA= 1.865 , SIGAF= 4.305 , SIGB= 1.16 , SIGC= 0.2811 NEUTRONS PER FISSION = 2.80. NZ=10. NPHI=10. FOR A SINGLE NEUTRON STRIKING THE SOURCE... THE TRANSMISSION PROBABILITY IS .18433302+0 THE INTERACTION PROBABILITY IS .81566698+0 THE ABSORPTION PROBABILITY IS .54509177+0 THE 'A' MATERIAL ABSORPTION INTERACTION PROBABILITY IS .11721783+0 THE «A' MATERIAL FISSION INTERACTION PROBABILITY IS .27057522+0 THE 'A« MATERIAL INTERACTION PROBABILITY IS .38779305+0

20 THE CLADDING MATERIAL INTERACTION PROBABILITY IS .42787393+0 NEUTRONS OUT PER NEUTRON STRIKE IS .94194362+0 NEUTRONS LOST PER NEUTRON STRIKE IS .58056380+0 THE AVERAGE CROSS SECTION OF THE SOURCE IN CM**2 IS .20170300+2 THE PERCENT ERROR IN THE TRANSMISSION PROBABILITY IS .0162 IF THE MEASURED THERMAL FLUX IN PERCENT OF Q PER CM**2 IS .122 THE INTERACTION IN PERCENT OF IS .20071742+1 THE XBSORPTION IN PERCENT OF Q IS .13413491+1 THE FISSION IN PERCENT OF Q IS .18643104+1 AND THE RESULTANT LOSS IN PERCENT OF Q IS .14286378+0

CYLINDRICAL SOURCE PU-BE A. ENCAPSULATED IN NICKEL. SOURCE DIMENSION INPUT IN INCHES, L = 1.031. OD = 1.031, B WALL = 0.128, B ENDS = 0.112 AND 0.129. MACROSCOPIC CROSS SECTIONS PER CM

SIGAA= 1.214 , SIGAF= 2.802 , SIGB= 0.4106. NEUTRONS PER FISSION = 2.80. NZ=15. NPHI=15. FOR A SINGLE NEUTRON STRIKING THE SOURCE... THE TRANSMISSION PROBABILITY IS .31458511+0 THE INTERACTION PROBABILITY IS .68541489+0 THE ABSORPTION PROBABILITY IS .38173937+0 THE «A» MATERIAL ABSORPTION INTERACTION PROBABILITY IS .13157105+0 THE »A' MATERIAL FISSION INTERACTION PROBABILITY IS .30367551+0 THE 'A' MATERIAL INTERACTION PROBABILITY IS .43524656+0 THE CLADDING MATERIAL INTERACTION PROBABILITY IS .25016833+0 NEUTRONS OUT PER NEUTRON STRIKE IS .11648765+1 NEUTRONS LOST PER NEUTRON STRIKE IS -.16487655+0 THE AVERAGE CROSS SECTION OF THE SOURCE IN CM**2 IS .77943026+1 THE PERCENT ERROR IN THE TRANSMISSION PROBABILITY IS .0270 IF THE MEASURED THERMAL FLUX IN PERCENT OF Q PER CM**2 IS .122 THE INTERACTION IN PERCENT OF Q IS .65176438+0 THE ABSORPTION IN PERCENT OF Q IS .36299785+0 THE FISSION IN PERCENT OF Q IS .80854631+0 AND THE RESULTANT LOSS IN PERCENT OF Q IS -.15678192+0 CYLINDRICAL AM-BE SOURCE. ENCAPSULATED IN TANTALUM AND STAINLESS STEEL. SOURCE DIMENSION INPUT IN INCHES, L = 1.355, OD = 1.355, B WALL = 0.07 B ENDS = 0.07, C WALL = 0.03, C ENDS = 0.05. MACROSCOPIC CROSS SECTIONS PER CM

SIGAA= .1704 , SIGAF= 0.00082 » SIGB= 1.16 , SIGC= 0.281 NEUTRONS PER FISSION = 2.89. NZ=10. NPHI=10.

FOR fi SINGLE NEUTRON STRIKING THE SOURCE... THE TRANSMISSION PROBABILITY IS .40678453+0 THE INTERACTION PROBABILITY IS .59321547+0 THE ABSORPTION PROBABILITY IS .59253429+0 THE «A' MATERIAL ABSORPTION INTERACTION PROBABILITY IS .14155335+0 THE «A' MATERIAL FISSION INTERACTION PROBABILITY IS .68118396-3 THE 'A' MATERIAL INTERACTION PROBABILITY IS .14223453+0 THE CLADDING MATERIAL INTERACTION PROBABILITY IS .45098094+0 NEUTRONS OUT PER NEUTRON STRIKE IS .40875315+0 NEUTRONS LOST PER NEUTRON STRIKE IS .59124685+0 THE AVERAGE CROSS SECTION OF THE SOURCE IN CM**2 IS .13463194+2 THE PERCENT ERROR IN THE TRANSMISSION PROBABILITY IS .0356 IF THE MEASURED THERMAL FLUX IN PERCENT OF Q PER CM**2 IS .121 THE INTERACTION IN PERCENT OF Q IS .96637561+0 THE ABSORPTION IN PERCENT OF Q IS .96526593+0 THE FISSION IN PERCENT OF Q IS .32069762-2 AND THE RESULTANT LOSS IN PERCENT OF Q IS .96316863+0

21 Table 1.

Neutron Transmission Convergence Test for the Spherical Source Program

NY Transmission Estimated Actual Error Error*

10 0.14653888 0.819$ 1.205$

20 0.14542886 0.438$ 0.438$

50 0.14495702 0.113$ 0.113$

100 0.14485061 0.041$ 0.039$

500 0.14479752 0.008$ 0.002$

1000 0.14479397 0.004$ —

* Actual Error listed here assumes that transmission for NY = 1000 is correct.

22 Y = x = o 1(a) VIEW FROM -Y 1(b) VIEW FROM +X

z=o z = z

1 (c) VIEW FROM +Z 1 (d) VIEW FROM +Z

Figure 1. Diagram for the cylindrical source program. The cylindrical source is surrounded by an imaginary sphere with center on the X axis in the middle of the core material. Figure 1(d) shows the intersection of a plane parallel to the X-Y axis with the source at Z.

23 0.3170

0.3160

O 0.3150

if) if)

if) 0.3140 < q: A-NZ=I0 0.3130 ?'

0.3120 4 8 12 16 NZ OR NPHI

Figure 2. The calculated Transmission and Estimated Error for a nickel encapsulated Pu-Be cylindrical source for various specified subdivisions of the Z and cp integrals, NZ and NPHI. The solid line joins points for whicl' NPHI equals 10 and the dashed line joins points for which NZ equals 10. Two points are shown for both NPHI and NZ equal to 5 and 15.

2k REFERENCES

1. Spiegel, Jr., V. and Murphey, W. M. : Metrologia (1971) (In press).

2. Murphey, W. M. : Nuclear Instr. Methods 37_, 13 (1965).

3. Mosburg, Jr., E. R. : J. Research Natl. Bur. Standards 62, 189 (1959).

4. Scarborough, J. B. : Numerical Mathematical Analysis , Baltimore: The Johns Hopkins Press: London: Humphrey Milford Oxford University Press 1930; pp. 120 and 155.

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7. AUTHOR(S) 8. Performing Organization V. Spiegel, Jr. and W. M. Murphey

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. Project/Task/ Work Unit No. 5030220 NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE 11. Contract/Grant No. WASHINGTON, D.C 20234

2. Sponsoring Organization Name and Address 13. Type of Report & Period Nuclear Radiation Division teal Institute for Basic Standards

National Bureau of Standards 14. Sponsoring Agency Code Washington, D.C. 20234

5. SUPPLEMENTARY NOTES

6. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) A computer code has been written in FORTRAN IV for the calculation of thermal neutron absorption in spherical and cylindrical neutron sources. The formalism of the calculation, the structure of the computer code, a listing of the code, and some sample results are presented. The com- parison of the results of this calculation to experiment appears in Metrologia 7, No. 1, 34-38 (1971).

17. KEY WORDS (Alphabetical order, separated by semicolons) Neutron; neutron standards; manganous sulfate bath calibration of neutron sources.

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